A $T_0$-space is a topological spaces $\mathbf{X}=\langle X,\Omega(\mathbf{X})\rangle$ such that

for every pair of distinct points in the space, there is an open set containing one but not the other: $x,y\in X\Longrightarrow\exists U\in\Omega(\mathbf{X})[(x\in U\mbox{ and }y\notin U)\mbox{ or }(y\in U\mbox{ and }x\notin U)]$